Hardy’s inequality
Suppose $p>1$ and $\{{a}_{n}\}$ is a sequence of nonnegative real numbers. Let ${A}_{n}={\sum}_{i=1}^{n}{a}_{i}$. Then
$$ |
unless all the ${a}_{n}$ are zero. The constant is best possible.
This theorem has an integral analogue: Suppose that $p>1$ and $f\ge 0$ on $(0,\mathrm{\infty})$. Let $F(x)={\int}_{0}^{x}f(t)\mathit{d}t$. Then
$$ |
unless $f\equiv 0$. The constant is best possible.
References
- 1 G.H. Hardy, J.E. Littlewood and G.Pólya, Inequalities^{}, Cambridge University Press, Cambridge, 2nd edition, 1952, pp. 239-240.
Title | Hardy’s inequality |
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Canonical name | HardysInequality |
Date of creation | 2013-03-22 17:04:32 |
Last modified on | 2013-03-22 17:04:32 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 8 |
Author | Mathprof (13753) |
Entry type | Theorem |
Classification | msc 26D15 |